Icy tornadoes in the quantum world - Josephson junctions of Bose-Einstein condensates : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematical Physics at Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University, Albany, New Zealand
This thesis presents a theoretical study of ultra-cold gases in, or close to a Bose- \nEinstein condensed phase. A system of two coupled Bose-Einstein condensates \nshows strong resemblance to Josephson junctions, consisting of two superconductors \nlinked via a thin insulating barrier. In these systems quantised ring \ncurrents across the barrier (Josephson vortices) have been detected. We concentrate \non similar macroscopic quantum states in our systems of linear and \nannular coupled Bose-Einstein condensates, and test them towards their potential \nfor showing macroscopic quantum tunnelling. \nFor the linear system we present a very detailed stability analysis of the stationary \nsolutions, the vortex and the soliton, using Bogoliubov-de Gennes theory. \nAn analytic approximation of the unstable mode is provided. We show that the \ntransition between vortices rotating in opposite directions is possible and propose \nan effective potential separating these two states. For the annular system \nof two coupled ring shaped condensates of different radii, we review the parameter \nregimes for finding vortices in the ground state. We show that pinning of \nvortices via a repulsive external potential is possible and suggest further steps \ntowards the detection of macroscopic quantum tunnelling.
This thesis presents a theoretical study of ultra-cold gases in, or close to a Bose- \nEinstein condensed phase. A system of two coupled Bose-Einstein condensates \nshows strong resemblance to Josephson junctions, consisting of two superconductors \nlinked via a thin insulating barrier. In these systems quantised ring \ncurrents across the barrier (Josephson vortices) have been detected. We concentrate \non similar macroscopic quantum states in our systems of linear and \nannular coupled Bose-Einstein condensates, and test them towards their potential \nfor showing macroscopic quantum tunnelling. \nFor the linear system we present a very detailed stability analysis of the stationary \nsolutions, the vortex and the soliton, using Bogoliubov-de Gennes theory. \nAn analytic approximation of the unstable mode is provided. We show that the \ntransition between vortices rotating in opposite directions is possible and propose \nan effective potential separating these two states. For the annular system \nof two coupled ring shaped condensates of different radii, we review the parameter \nregimes for finding vortices in the ground state. We show that pinning of \nvortices via a repulsive external potential is possible and suggest further steps \ntowards the detection of macroscopic quantum tunnelling.
This thesis presents a theoretical study of Bose-Einstein condensates in a doublering trap. In particular, we determine the ground states of the condensate in the double-ring trap that arise from the interplay of quantum tunnelling and the trap’s rotation. The trap geometry is a concentric ring system, where the inner ring is of smaller radius than the outer ring and both lie in the same two-dimensional plane. Due to the difference in radii between the inner and outer rings, the angular momentum that minimises the kinetic energy of a condensate when confined in the individual rings is different at most frequencies. This preference is in direct competition with the tunnel coupling of the rings which favours the same angular momentum states being occupied in both rings. Our calculations show that at low tunnel coupling ground state solutions exist where the expectation value of angular momentum per atom in each ring differs by approximately an integer multiple. The energy of these solutions is minimised by maintaining a uniform phase difference around most of the ring, and introducing a Josephson vortex between the inner and outer rings. A Josephson vortex is identified by a 2p step in the relative phase between the two rings, and accounts for one quantum of circulation. We discuss similarities and differences between Josephson vortices in cold-atom systems and in superconducting Josephson junctions. Josephson vortices are actuated by a sudden change in the trapping potential. After this change Josephson vortices rotate around the double-ring system at a different frequency to the rotation of the double-ring potential. Numerical studies of the dependence of the velocity on the ground state tunnel coupling and interaction strength are presented. An analytical theory of the Josephson vortex dynamics is also presented which is consistent with our numerical results.
In this thesis, we will study the superfluidity of condensed atomic and photonic systems, through the manipulation of rotational states, such as vortices or persistent currents. We will study Bose-Einstein condensates both in the strongly-correlated regime, where models based on second quantization, like Bose-Hubbard model, will be required; and in weakly-interacting systems, where mean-field approximations will be accurate enough, and the system is described by means of the Gross-Pitaevskii equation.\n\n\t\t\t\t We will start with the analysis of the fundamental properties of Bose gases trapped in few-site lattices, such as the phase diagram, the condensed fractions and the entanglement. Concerning the phases, we will study the properties of the transitions between them and, in particular, their characteristic critical exponents.\n\n\t\t\t\t Afterwards, we will consider the sites of a lattice constituting a ring geometry and study the effect of manipulating the tunnelling rate between two of the wells. This kind of tunable link is called weak link, and we will analyze what happens in the mean-field approximation, in comparison with the strongly-correlated case. In both regimes we will observe that the weak link behaves as a key element in the system in order to generate superpositions of flow states. Moreover, in the mean-field case, we can identify an energy barrier that separates two current states (also known as winding number states), where solitonic states, i.e. states characterized by the presence of topological singularities, live. Such a barrier will be the origin of the appearance of a hysteresis cicle in processes of transfer between different winding number curves, called phase slips.\n\n\t\t\t\t After that, we will study two coherently-coupled components of a toroidally-trapped Bose-Einstein condensate. We will see that when we imprint a persistent current in one of the components, there is an angular momentum transfer between both components. This transfer can be identified as a phase slip event, and the tunability of the system allows it to behave as a robust qubit, due to the fact that states supported by currents are less fragile.\n\n\t\t\t\t In two-component condensates, it is possible to find a particular solitonic state called Josephson vortex. This state is characterized by a density depletion around a point with nonzero currents. Moreover, these states are energetically more favourable than dark soliton states, whose main difference with respect to Josephson vortices is the fact that dark solitons do not present currents. However, when spin-orbit coupling is added, dark soliton states are no longer possible, but Josephson vortices persist. In this thesis, we will see that these states decay through transversal excitations (i.e. snake instability), producing vortex-antivortex pairs, and their subsequent dynamical evolution depends on the initial orientation of the Josephson vortex.\n\n\t\t\t\t Finally, we will move to the field of polariton condensates. Polaritons are quasiparticles product of the coupling between photons and excitons (which are electron-hole excitations) in semiconductor microcavities. These particles can constitute an out-of-equilibrium (due to the short lifetime of polaritons) Bose-Einstein condensate described by the Gross-Pitaevskii-like equation for two components, because of the two polarization components inherent to the photonic nature of polaritons. The cavities where these condensates are created generate a spin-orbit coupling between the two polariton components, in such a way that current states with different orbital angular momentum are coupled. This yields to a phenomenon of spin-to-orbital angular momentum conversion that we will study in ring-trapped polariton condensates. At the end of this thesis, we will probe the superfluid properties of polariton condensates, by analyzing the response of the generated currents against the presence of disorder.
Bose-Einstein condensation is a quantum statistical phase transition which was theoretically predicted almost a hundred years ago. After years of seminal research, physicists realized the first almost ideal Bose-Einstein condensates in ultracold dilute atomic gases in 1995. Since then, the theoretical and experimental methods concerning such systems have been developing rapidly, and many fascinating phenomena have been found in these novel quantum systems.\n\nBose-Einstein condensation occurs in a system consisting of massive bosons when a single quantum state becomes macroscopically occupied as the temperature is lowered below the transition temperature. In general, condensates consisting of repulsively interacting bosons exhibit superfluidity: Particle currents can flow in the system without dissipation and viscosity. Moreover, the velocity fields of condensates have to be irrotational, which severely restricts the rotational characteristics of these systems. Apart from the center of mass motion, the system may carry angular momentum in the form of elementary excitations or so-called quantized vortices.\n\nThis Thesis is a theoretical study of subjects related to stability and dynamics of quantized vortices in dilute atomic Bose-Einstein condensates. The precession and instability of off-centered vortices in trapped condensates is investigated both in the zero-temperature limit and at finite temperatures. Dynamical stability of multiply quantized vortices and vortex clusters is studied in axisymmetric trap geometries. Splitting of energetically and dynamically unstable multiply quantized vortices into singly quantized vortices is also studied. Finally, as a separate subject, tunneling of a condensate through a potential barrier is investigated. Majority of this work relies on numerical methods for solving the Gross-Pitaevskii and Bogoliubov equations, which are of central importance in the study of dilute atomic Bose-Einstein condensates.
Since the first realizations of Bose-Einstein condensation in dilute alkali-atom vapours in 1995, these novel quantum gases have attracted wide interest. The main focus of the research has been on finding out the coherence and superfluidity properties of the condensates. From the theoretical point of view, these systems provide a rare possibility to model an interacting many-particle system exhaustively and quantitatively accurately from first-principles quantum field theories. However, the challenge of developing a correct and computationally feasible finite-temperature formalism still remains.\n\nThe coherence and superfluidity properties are intimately related to the existence and stability of quantized vortices. In this thesis, the structure and stability of vortex states in weakly interacting Bose-Einstein condensates is investigated within microscopic finite-temperature mean-field theories. An effective scheme for computationally solving the Bogoliubov-de Gennes quasiparticle eigenequations is developed, and applied to study the microscopic structure of vortex states in partially condensed vapours. In contrast to experimental observations and the predictions of the zero-temperature Bogoliubov approximation, stationary vortex states are found to be metastable within the self-consistent Popov approximation and its extensions even in the zero-temperature limit.\n\nThe reasons underlying this discrepancy are analysed in detail. A criterion for the validity of stationary mean-field approximations in modelling slowly time-dependent systems is derived, and applied to precessing off-axis vortex states. Under typical experimental conditions moving vortices are shown to be deformed due to the inability of the thermal cloud to follow the vortex core rigidly. Consequently, the validity of the commonly used stationary mean-field theories in describing such states is shown to be questionable. Finally, an analysis of uniformly precessing off-axis vortex states within a fully time-dependent Popov approximation is presented. Even dynamically deformed vortices are shown to become energetically metastable as the vortex core is partially filled with noncondensed gas in the course of thermalization. This self-stabilization mechanism of vortex states, shown to occur under very general circumstances, is an inherent property of all self-consistent formalisms having a structure similar to the Popov approximation. However, it is argued to be affected by the improper treatment of the noncondensate dynamics within these formalisms, and the need for a theory taking properly into account the coupled dynamics of the condensate and the noncondensate in modelling vortex states is emphasized. The computational scheme developed is also applied to investigate possibilities to energetically stabilize multiply quantized vortex states by piercing an externally rotated condensate with a blue-detuned laser beam.
Bose-Einstein condensation, first predicted by Bose and Einstein in 1924, was finally realized in dilute, weakly interacting gases of alkali atoms in 1995. A few years later, quantized vortices - intimately related to the phenomenon of superfluidity - were created and observed in these novel quantum many-body systems.\n\nIn this Thesis, vortices in dilute atomic Bose-Einstein condensates are studied theoretically and computationally using microscopic mean-field theory formalisms. Special emphasis is on the structure and the stability of vortices at finite temperatures. The formation of quantized circulation in Bose-Einstein condensates is investigated, as well as multiply quantized vorticity and the adiabaticity of the precessing motion of a vortex.\n\nThis research strongly relies on numerically solving the Gross-Pitaevskii equation for the macroscopic condensate wavefunction and the coupled Bogoliubov equations and their variants for the collective quasiparticle excitation eigenmodes. The computed elementary excitation spectra are used, for instance, in determining the local stability of various condensate states. The vortex formation process is studied at finite temperatures in terms of the surface modes of the condensate providing the Landau critical velocities for the breakdown of superfluidity. Free energy calculations are used for determining the thermodynamic equilibrium states.\n\nSingly quantized, axisymmetric vortex states are found to be locally energetically stable even in the zero-temperature limit within the self-consistent finite-temperature approximations. This would imply for the vortex a precession direction opposite to that observed in the experiment. Adiabatic approximation of quantum mechanics is formulated for Bose-Einstein condensed systems and is applied to a case of a moving vortex line. The experimentally observed precessing motion of the vortex is found to be nonadiabatic. A method to create stable multiply quantized vortices using an additional pinning potential is presented. The computed critical trap rotation frequencies for vortex nucleation at finite temperatures are in agreement with experiments and the predictions of zero-temperature field theories.
The main objective of this thesis is to explain, from the quantum-mechanical point of view, the nature of dark solitons in one-dimensional cold-atom systems. Models of bosons and fermions with contact interactions on a ring are exactly solvable via the Bethe ansatz, and support so-called type-II elementary excitations. These have long been associated with dark solitons of the Gross-Pitaevskii equation due to the similarity of the dispersion relation, despite the completely different physical properties of the states. Fully understanding this connection is our primary aim. We begin by reviewing the Gross-Pitaevskii equation and its dark soliton solutions. Next, we solve the mean-field problem of two coupled one-dimensional Bose-Einstein condensates, with special emphasis on Josephson vortices and their dispersion relation. Predictions are given for possible experimental detection. Then we give a derivation that justifies a method for the extraction of the so-called missing particle number from the dispersion relation of solitonic excitations. A derivation of the finite Bethe ansatz equations for the Lieb-Liniger and Yang-Gaudin models follows. These describe a single species of bosons and two component fermions, respectively. We review the elementary excitations of the Lieb-Linger model, and carry out a comprehensive study of the (much richer) excitations of the Yang-Gaudin model. The thermodynamic limit Bethe ansatz equations for all states of interest in both models are derived, and the 𝘮𝘪𝘴𝘴𝘪𝘯𝘨 𝘱𝘢𝘳𝘵𝘪𝘤𝘭𝘦 𝘯𝘶𝘮𝘣𝘦𝘳 and the closely-related 𝘱𝘩𝘢𝘴𝘦-𝘴𝘵𝘦𝘱 are extracted from the dispersion relations. Next, we develop a method for approximating the finite-system dispersion relation of solitonic excitations from the thermodynamic limit results. Finally, we show that the single particle density and phase profiles of appropriately formed superpositions of type-II states with different momenta exhibit solitonic features. Through this idea, the missing particle number and phase step extracted from the dispersion relation gain physical meaning. Moreover, we use a convolution model to extract the fundamental quantum dark soliton length scale across the range of interactions and momenta. The insight gained in the bosonic case is used to make inferences about dark solitons in the fermionic case. Furthermore, we study the Hess-Fairbank effect in the repulsive Yang-Gaudin model and the fermionic super Tonks-Girardeau regime.